Problem Solving Dr. Andrew Wallace PhD BEng(hons) EurIng

Overview Breaking and entry How to get away? Which items to steal? Stash the loot

Brute Force Simple Start at the beginning Keep working till.. You find a solution! Bang you head against a brick wall long enough, sooner or later the brick wall gives out

Brute Force

Searching a list Travelling salesman problem Deoxyribonucleic acid Heuristics!

Brute Force It can work! Some times it’s the only solution! Problem: Inefficient Can become time consuming

Optimal subproblems Optimal solution has optimal subproblem solutions S = {s 1, s 2, s 3, … s n } Optimal sub-structure Greedy algorithms Else Overlapping substructures Dynamic programming

Greedy Algorithms Local optimum results in global optimum Problem: Shortest path from A to B Museum Safe House

Greedy Algorithms Not all local optimal solutions can lead to global optimal solutions Can end up with the worse case If globally optimal can be fast

Greedy Algorithms Set of candidate solutions Paths Selection function Which path to choose? End function Are we there yet?

Greedy Algorithms A resource Set of agents who want to use the resource No overlaps in time allowed Agents (a)abcdefg Start (s) Finish (f) a, d, e

Greedy Algorithms Earliest finishing time to maximise the number of participants Maximise the remaining time

Greedy Algorithms O(n)

Greedy Algorithms No back tracking Choices dependent on passed choices but not future choices

Dynamic programming Overlapping sub problems Once calculated, save and reuse Dynamic Updates and changes things but not as in dynamic programming languages Programming Filling tables not computer programming

Dynamic programming Items Weight Value But we have … Max weight!

Dynamic programming Fibonacci sequence F n = F n-1 + F n-2 F 0 = 0, F 1 = 1 and F 2 = 1 F(6) F(4) F(5) F(3) F(4) F(3) F(1)F(2) F(0)F(1) F(2) F(0) F(1) F(2) F(0)F(1) F(2) F(0)F(1) F(2) F(1)F(0) F(1)

Dynamic programming Items = {i 1, i 2, i 3 … i n } Weight = {w 1, w 2, w 3 … w n } Value = {v 1, v 2, v 3 … v n } W = max weight we can take

Dynamic programming Optimal structure? w = {w 1, w 2 … w j } w = {w 1, w 2 … w j-1 } Overlapping subproblems Compare solution with item to solution without item

Dynamic programming If i = 0 or w= 0 If w i > w If i > 0 or w >= w i

Dynamic programming O(n)

Divide and Conquer Recursion Break the problem reclusively into small problems Preferably evenly Solve the smaller problems Combine together to produce the overall solution

Divide and Conquer T(n) = 2T(n/2) +  (n) Master method

Divide and Conquer

Find

Divide and Conquer Find 90 Divide into two

Divide and Conquer Find 90 Divide into two And again!

Divide and Conquer Find 90 Divide into two And again! 90

Divide and Conquer

Problem solving Top down greedy Bottom up dynamic

Questions?